Continuous modeling of grain size segregation in bedload transport

2020 ◽  
Author(s):  
Hugo Rousseau ◽  
Rémi Chassagne ◽  
Julien Chauchat ◽  
Philippe Frey
Keyword(s):  
Grain Size ◽  
Mixture Theory ◽  
Bedload Transport ◽  
Force Balance ◽  
Size Segregation ◽  
Small Particles ◽  
Class Model ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
And Diffusion

<p>Rivers carry sediments having a wide grain size distribution, ranging from a few hundreds microns to meters. This leads to grain size segregation mechanism that can have huge consequences on morphological evolution.  Accurate comprehension and modeling of this mechanism with continuous equations is a key step to upscale segregation in sediment transport models.<br><em>Thornton et al. (2006) </em>developed continuous equations for bidisperse segregation in the context of the mixture theory. Based on the momentum balance of small particles, a simple advection-diffusion equation for the volumetric concentration of small particles was derived. This equation enables to explicit the advection term, that tends to segregate the different particle sizes and the diffusive term, that tends to remix the particles. However, this approach does not immediately provide the physical characteristics of the granular flow in the advection and diffusion terms.</p><p>Recently, <em>Guillard et al. (2016)</em> showed, using a Discrete Element Method (DEM), that the segregation force on a large intruder in a bath of small particles, can be seen as a buoyancy force proportional to the pressure. In addition, <em>Tripathi and Khakhar (2011)</em> showed that a large particle rising in a pool of small grains experiences a Stokesian drag force proportional to the granular viscosity.<br>These new results enable to infer a force balance for a single coarse particle in bedload transport. Solving this force balance showed that the large particle rises with the accurate dynamics, meaning that this force balance is relevant to model grain-size segregation.</p><p>Based on these new forces, a continuous multi-class model has been developed to generalize to the segregation of a collection of large particles. The concentration and the segregation velocity of the small particles have been compared with coupled-fluid DEM bedload transport simulations from <em>Chassagne et al. (2020)</em> and show that the accurate dynamics of segregation can be modeled using this continuous model.<br>Based on this continuum multi-class model, a similar advection-diffusion equation as <em>Thornton et al. (2006) </em> has been obtained. The latter appears to provide the physical origin of the advection and diffusion terms by linking them to the parameters of the flow.</p><p> </p><p>Chassagne R., Maurin R., Chauchat J., and Frey P. Discrete and continuum modeling of grain-size segregation during bedload transport. J. Fluid Mech. 2020 (in revision).</p><p>Gray J. M. N. T., and  Chugunov V. A. Particle-size segregation and diffusive remixing in shallow granular avalanches. J. Fluid Mech. 569: 365-398, 2006.</p><p>Guillard F. Forterre Y., and Pouliquen O. Scaling laws for segregation forces in dense sheared granular flows. J. Fluid Mech. 807, R1, 2016.</p><p>Thornton A. R., Gray J. M. N. T., and Hogg A. J. A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid Mech. 550: 125, 2006.</p><p>Tripathi A., and Khakhar D. V. Numerical simulation of the sedimentation of a sphere in a sheared granular fluid: a granular stokes experiment. Phys. Rev. Lett. 107, 108,001, 2011.</p>

scholarly journals On FTCS Approach for Box Model of Three-Dimension Advection-Diffusion Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Jeffry Kusuma ◽  
Agustinus Ribal ◽  
Andi Galsan Mahie
Keyword(s):  
Mathematical Model ◽  
Three Dimension ◽  
Sand Mining ◽  
Mining Pollution ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Diffusion Parameters ◽  
Sea Sand ◽  
And Diffusion ◽  
Pollution Distribution

This paper describes a numerical solution for mathematical model of the transport equation in a simple rectangular box domain. The model of street tunnel pollution distribution using two-dimension advection and three-dimension diffusion is solved numerically. Because of the nature of the problem, the model is extended to become three-dimension advection and three-dimension diffusion to study the sea-sand mining pollution distribution. This model with various advection and diffusion parameters and the boundaries conditions is also solved numerically using a finite difference (FTCS) method.

scholarly journals Discrete and continuum modelling of grain size segregation during bedload transport

2020 ◽  
Vol 895 ◽  
Author(s):  
Rémi Chassagne ◽  
Raphaël Maurin ◽  
Julien Chauchat ◽  
J. M. N. T. Gray ◽  
Philippe Frey
Keyword(s):  
Grain Size ◽  
Bedload Transport ◽  
Size Segregation ◽  
Continuum Modelling ◽  
Image Position

scholarly journals Numerical solution of advection-diffusion equation using finite difference schemes

2020 ◽  
Vol 55 (1) ◽  
pp. 15-22
Author(s):  
LS Andallah ◽  
MR Khatun
Keyword(s):  
Diffusion Equation ◽  
Numerical Solution ◽  
Numerical Scheme ◽  
Finite Difference Schemes ◽  
Qualitative Behavior ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Nicolson Scheme ◽  
And Diffusion ◽  
Crank Nicolson

This paper presents numerical simulation of one-dimensional advection-diffusion equation. We study the analytical solution of advection diffusion equation as an initial value problem in infinite space and realize the qualitative behavior of the solution in terms of advection and diffusion co-efficient. We obtain the numerical solution of this equation by using explicit centered difference scheme and Crank-Nicolson scheme for prescribed initial and boundary data. We implement the numerical scheme by developing a computer programming code and present the stability analysis of Crank-Nicolson scheme for ADE. For the validity test, we perform error estimation of the numerical scheme and presented the numerical features of rate of convergence graphically. The qualitative behavior of the ADE for different choice of the advection and diffusion co-efficient is verified. Finally, we estimate the pollutant in a river at different times and different points by using these numerical scheme. Bangladesh J. Sci. Ind. Res.55(1), 15-22, 2020

A COMPARISON OF SPLINES INTERPOLATIONS WITH STANDARD FINITE DIFFERENCE METHODS FOR ONE-DIMENSIONAL ADVECTION-DIFFUSION EQUATION

2008 ◽  
Vol 19 (08) ◽  
pp. 1291-1304 ◽  
Author(s):  
MONTRI THONGMOON ◽  
SUWON TANGMANEE ◽  
ROBERT MCKIBBIN
Keyword(s):  
Finite Difference Methods ◽  
Cubic Spline ◽  
One Dimensional ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Constant Coefficients ◽  
Difference Methods ◽  
The One ◽  
Natural Cubic Spline ◽  
And Diffusion

Four types of numerical methods namely: Natural Cubic Spline, Special A-D Cubic Spline, FTCS and Crank–Nicolson are applied to both advection and diffusion terms of the one-dimensional advection-diffusion equations with constant coefficients. The numerical results from two examples are tested with the known analytical solution. The errors are compared when using different Peclet numbers.

scholarly journals Multi-component particle-size segregation in shallow granular avalanches

2011 ◽  
Vol 678 ◽  
pp. 535-588 ◽  
Author(s):  
J. M. N. T. GRAY ◽  
C. ANCEY
Keyword(s):  
Particle Size ◽  
Grain Size ◽  
Ternary Mixture ◽  
Numerical Solutions ◽  
Time Dependent ◽  
Size Segregation ◽  
Small Particles ◽  
Near Surface ◽  
Fine Grained ◽  
Fine Grained Material

A general continuum theory for particle-size segregation and diffusive remixing in polydisperse granular avalanches is formulated using mixture theory. Comparisons are drawn to existing segregation theories for bi-disperse mixtures and the case of a ternary mixture of large, medium and small particles is investigated. In this case, the general theory reduces to a system of two coupled parabolic segregation–remixing equations, which have a single diffusion coefficient and three parameters which control the segregation rates between each pair of constituents. Considerable insight into many problems where the effect of diffusive remixing is small is provided by the non-diffusive case. Here the equations reduce to a system of two first-order conservation laws, whose wave speeds are real for a very wide class of segregation parameters. In this regime, the system is guaranteed to be non-strictly hyperbolic for all admissible concentrations. If the segregation rates do not increase monotonically with the grain-size ratio, it is possible to enter another region of parameter space, where the equations may either be hyperbolic or elliptic, depending on the segregation rates and the local particle concentrations. Even if the solution is initially hyperbolic everywhere, regions of ellipticity may develop during the evolution of the problem. Such regions in a time-dependent problem necessarily lead to short wavelength Hadamard instability and ill-posedness. A linear stability analysis is used to show that the diffusive remixing terms are sufficient to regularize the theory and prevent unbounded growth rates at high wave numbers. Numerical solutions for the time-dependent segregation of an initially almost homogeneously mixed state are performed using a standard Galerkin finite element method. The diffuse solutions may be linearly stable or unstable, depending on the initial concentrations. In the linearly unstable region, ‘sawtooth’ concentration stripes form that trap and focus the medium-sized grains. The large and small particles still percolate through the avalanche and separate out at the surface and base of the flow due to the no-flux boundary conditions. As these regions grow, the unstable striped region is annihilated. The theory is used to investigate inverse distribution grading and reverse coarse-tail grading in multi-component mixtures. These terms are commonly used by geologists to describe particle-size distributions in which either the whole grain-size population coarsens upwards, or just the coarsest clasts are inversely graded and a fine-grained matrix is found everywhere. An exact solution is constructed for the steady segregation of a ternary mixture as it flows down an inclined slope from an initially homogeneously mixed inflow. It shows that for distribution grading, the particles segregate out into three inversely graded sharply segregated layers sufficiently far downstream, with the largest particles on top, the fines at the bottom and the medium-sized grains sandwiched in between. The heights of the layers are strongly influenced by the downstream velocity profile, with layers becoming thinner in the faster moving near-surface regions of the avalanche, and thicker in the slowly moving basal layers, for the same mass flux. Conditions for the existence of the solution are discussed and a simple and useful upper bound is derived for the distance at which all the particles completely segregate. When the effects of diffusive remixing are included, the sharp concentration discontinuities are smoothed out, but the simple shock solutions capture many features of the evolving size distribution for typical diffusive remixing rates. The theory is also used to construct a simple model for reverse coarse-tail grading, in which the fine-grained material does not segregate. The numerical method is used to calculate diffuse solutions for a ternary mixture and a sharply segregated shock solution is derived that looks similar to the segregation of a bi-disperse mixture of large and medium grains. The presence of the fine-grained material, however, prevents high concentrations of large or medium particles being achieved and there is a significant lengthening of the segregation distance.

scholarly journals A theory for particle size segregation in shallow granular free-surface flows

2005 ◽  
Vol 461 (2057) ◽  
pp. 1447-1473 ◽  
Author(s):  
J.M.N.T Gray ◽  
A.R Thornton
Keyword(s):  
Free Surface ◽  
Plug Flow ◽  
Mixture Theory ◽  
Free Surface Flows ◽  
Size Segregation ◽  
Void Space ◽  
Small Particles ◽  
Surface Flows ◽  
Large Particles ◽  
Steady State Solutions

Abstract Granular materials composed of a mixture of grain sizes are notoriously prone to segregation during shaking or transport. In this paper, a binary mixture theory is used to formulate a model for kinetic sieving of large and small particles in thin, rapidly flowing avalanches, which occur in many industrial and geophysical free-surface flows. The model is based on a simple percolation idea, in which the small particles preferentially fall into underlying void space and lever large particles upwards. Exact steady-state solutions have been constructed for general steady uniform velocity fields, as well as time-dependent solutions for plug-flow, that exploit the decoupling of material columns in the avalanche. All the solutions indicate the development of concentration shocks, which are frequently observed in experiments. A shock-capturing numerical algorithm is formulated to solve general problems and is used to investigate segregation in flows with weak shear.

scholarly journals Experiments on grain size segregation in bedload transport on a steep slope

2020 ◽  
Vol 136 ◽  
pp. 103478
Author(s):  
P. Frey ◽  
H. Lafaye de Micheaux ◽  
C. Bel ◽  
R. Maurin ◽  
K. Rorsman ◽  
...  
Keyword(s):  
Grain Size ◽  
Bedload Transport ◽  
Steep Slope ◽  
Size Segregation

scholarly journals Global heat balance and heat uptake in potential temperature coordinates

2021 ◽  
Author(s):  
Antoine Hochet ◽  
Rémi Tailleux ◽  
Till Kuhlbrodt ◽  
David Ferreira
Keyword(s):  
Global Warming ◽  
Water Mass ◽  
Potential Temperature ◽  
Effective Diffusion ◽  
Mitigation Strategies ◽  
Ocean Heat Uptake ◽  
Cold Temperatures ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Heat Uptake

AbstractThe representation of ocean heat uptake in Simple Climate Models used for policy advice on climate change mitigation strategies is often based on variants of the one-dimensional Vertical Advection/Diffusion equation (VAD) for some averaged form of potential temperature. In such models, the effective advection and turbulent diffusion are usually tuned to emulate the behaviour of a given target climate model. However, because the statistical nature of such a “behavioural” calibration usually obscures the exact dependence of the effective diffusion and advection on the actual physical processes responsible for ocean heat uptake, it is difficult to understand its limitations and how to go about improving VADs. This paper proposes a physical calibration of the VAD that aims to provide explicit traceability of effective diffusion and advection to the processes responsible for ocean heat uptake. This construction relies on the coarse-graining of the full three-dimensional advection diffusion for potential temperature using potential temperature coordinates. The main advantage of this formulation is that the temporal evolution of the reference temperature profile is entirely due to the competition between effective diffusivity that is always positive definite, and the water mass transformation taking place at the surface, as in classical water mass analyses literature. These quantities are evaluated in numerical simulations of present day climate and global warming experiments. In this framework, the heat uptake in the global warming experiment is attributed to the increase of surface heat flux at low latitudes, its decrease at high latitudes and to the redistribution of heat toward cold temperatures made by diffusive flux.

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